On the eigenvalues of distance powers of circuits

TL;DR

This paper investigates the eigenvalues of distance powers of circuits, establishing which integers can occur as eigenvalues and providing explicit eigenspace bases with vectors having entries -1, 0, 1.

Contribution

It characterizes the integer eigenvalues of circuit distance powers, showing they are limited to -3 to 2d, and constructs explicit eigenspaces with simple vectors.

Findings

Only -3, -2, -1, 0, 1, 2d can be integer eigenvalues.

Eigenvalue multiplicities are explicitly determined.

Explicit eigenspace bases with vectors in {-1, 0, 1} are provided.

Abstract

Taking the d-th distance power of a graph, one adds edges between all pairs of vertices of that graph whose distance is at most d. It is shown that only the numbers -3, -2, -1, 0, 1, 2d can be integer eigenvalues of a circuit distance power. Moreover, their respective multiplicities are determined and explicit constructions for corresponding eigenspace bases containing only vectors with entries -1, 0, 1 are given.